#1761

Hard

LeetCode Problem Workspace

Minimum Degree of a Connected Trio in a Graph

Find the minimum degree of a connected trio in a graph using enumeration over nodes and edges.

Problem Statement

You are given an undirected graph with n nodes and a list of edges. Each edge connects two nodes in the graph, and no edges are repeated. A trio of nodes forms a connected trio if every pair of nodes in the trio has an edge between them.

The degree of a connected trio is defined as the number of edges where one endpoint is part of the trio and the other endpoint is not. Your task is to find the minimum degree of any connected trio in the graph.

Examples

Example 1

Input: n = 6, edges = [[1,2],[1,3],[3,2],[4,1],[5,2],[3,6]]

Output: 3

There is exactly one trio, which is [1,2,3]. The edges that form its degree are bolded in the figure above.

Example 2

Input: n = 7, edges = [[1,3],[4,1],[4,3],[2,5],[5,6],[6,7],[7,5],[2,6]]

Output: 0

There are exactly three trios:

[1,4,3] with degree 0.
[2,5,6] with degree 2.
[5,6,7] with degree 2.

Constraints

2 <= n <= 400
edges[i].length == 2
1 <= edges.length <= n * (n-1) / 2
1 <= ui, vi <= n
ui != vi
There are no repeated edges.

Solution Approach

Naive Enumeration

You can try enumerating all possible trios and then check if they form a connected trio. For each trio, compute its degree by checking how many edges are involved with the trio. This approach will give a correct answer but might not be efficient for larger graphs.

Optimized Graph Traversal

By leveraging graph traversal techniques like BFS or DFS, you can efficiently explore the neighbors of each node and determine the connected trios. This avoids checking every possible combination directly, reducing the overall complexity.

Degree Counting with Reduced Search Space

Consider reducing the search space by focusing on nodes with higher degrees first. Since the degree of the trio is directly related to the degrees of the nodes involved, prioritizing nodes with a higher degree could lead to faster identification of the minimum degree connected trio.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the chosen approach. The naive enumeration approach has a complexity of O(n^3) due to checking every trio. Optimized approaches using graph traversal or degree counting may improve this to O(n^2) or better, depending on implementation details. Space complexity is typically O(n) for graph representation and auxiliary structures.

What Interviewers Usually Probe

Understanding of graph enumeration patterns and efficient counting techniques.
Ability to apply traversal strategies for reducing unnecessary computations.
Comfort with optimizing brute force solutions to scale to larger inputs.

Common Pitfalls or Variants

Common pitfalls

Incorrectly counting the degree of a trio by missing some edges between nodes.
Failing to properly check all connected trios, especially in sparse graphs.
Using brute force methods without optimization that could lead to excessive runtime for larger graphs.

Follow-up variants

Find the maximum degree of a connected trio.
Handle directed graphs where the degree of a trio might change based on edge direction.
Modify the problem to allow for weighted edges and find the minimum weighted degree of a connected trio.

FAQ

How do I solve the Minimum Degree of a Connected Trio in a Graph problem efficiently?

Use graph traversal techniques like BFS or DFS to reduce unnecessary checking of all node combinations. Prioritize nodes with higher degrees for better performance.

What is the key pattern in the Minimum Degree of a Connected Trio in a Graph problem?

The key pattern is using graph enumeration combined with degree counting to identify the trio with the minimum degree efficiently.

How do I calculate the degree of a connected trio?

The degree of a connected trio is calculated by counting the edges where one endpoint is part of the trio and the other is not. The degree is the sum of the individual degrees minus 6.

Can this problem be solved with a greedy approach?

A greedy approach might not always work, as it may not guarantee finding the optimal solution. Efficient graph enumeration with careful counting is the best approach.

What should I avoid when solving the Minimum Degree of a Connected Trio problem?

Avoid brute-force checking of all trios without optimization. Also, make sure to correctly count the edges and ensure that all nodes in the trio are connected.

terminal

Solution

Solution 1: Brute Force Enumeration

We first store all edges in the adjacency matrix $\textit{g}$, and then store the degree of each node in the array $\textit{deg}$. Initialize the answer $\textit{ans} = +\infty$.

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def min(a: int, b: int) -> int:
    return a if a < b else b


class Solution:
    def minTrioDegree(self, n: int, edges: List[List[int]]) -> int:
        g = [[False] * n for _ in range(n)]
        deg = [0] * n
        for u, v in edges:
            u, v = u - 1, v - 1
            g[u][v] = g[v][u] = True
            deg[u] += 1
            deg[v] += 1
        ans = inf
        for i in range(n):
            for j in range(i + 1, n):
                if g[i][j]:
                    for k in range(j + 1, n):
                        if g[i][k] and g[j][k]:
                            ans = min(ans, deg[i] + deg[j] + deg[k] - 6)
        return -1 if ans == inf else ans

Continue Practicing

#2242 maximum score of a node sequence #1782 count pairs of nodes #1786 number of restricted paths from first to last node